Absolutely! Let's walk through the problem **step by step** and discuss **significant figures** at each stage. --- ## 1. **Calculate the Value of y** Given: - \( a = 5. \, \text{m} \) - \( b = 2.22 \, \text{m} \) - \( c = 3.6 \, \text{m} \) - Equation: \( y = a + b - c \) So: \[ y = 5. + 2.22 - 3.6 = 3.62 \, \text{m} \] **Significant Figures:** For addition/subtraction, the answer must have the same number of decimal places as the value with the **least decimal places**. - \( 5. \) (1 decimal) - \( 2.22 \) (2 decimals) - \( 3.6 \) (1 decimal) So, round \( y \) to **1 decimal place**: \[ y = 3.6 \, \text{m} \] --- ## 2. **Calculate the Propagated Error (\( s_y \))** The formula (root sum of squares): \[ s_y = \sqrt{s_a^2 + s_b^2 + s_c^2} \] Where: - \( s_a = .2 \, \text{m} \) - \( s_b = .07 \, \text{m} \) - \( s_c = .1 \, \text{m} \) Plug in: \[ s_y = \sqrt{(.2)^2 + (.07)^2 + (.1)^2} \] \[ s_y = \sqrt{.04 + .0049 + .01} \] \[ s_y = \sqrt{.0549} \] \[ s_y \approx .234 \, \text{m} \] **Significant Figures for Error:** Errors should be reported with **1 significant figure** (sometimes 2 if the first digit is 1 or 2 for clarity), so: \[ s_y \approx .2 \, \text{m} \] --- ## 3. **Final Answer (Value ± Error)** - Value: \( 3.6 \, \text{m} \) - Error: \( .2 \, \text{m} \) **Format:** \( y = 3.6 \pm .2 \, \text{m} \) - The answer is rounded to the decimal place of the error (tenths place). --- ## 4. **Relative Error** Relative error is: \[ \text{Relative error} = \frac{s_y}{y} \] \[ \frac{.2}{3.6} \approx .056 \] **Significant Figures:** Relative errors are often given with **1 or 2 significant figures**: \[ \text{Relative error} = .06 \pm .02 \] But since the error is already 1 significant figure: \[ \boxed{.06 \pm .02} \] --- ## 5. **Percent Relative Error** Percent relative error: \[ \text{Percent relative error} = \left(\frac{s_y}{y}\right) \times 100\% \] \[ = .056 \times 100\% \approx 5.6\% \] **Significant Figures:** Match the significant figures of the relative error: \[ \boxed{6\%} \] --- ## **Summary Table** | Step | Value/Formula | Result | Significant Figures | |-----------------------------|----------------------------------|-----------------------|-------------------------------| | Final answer (value ± error)| \( 3.6 \pm .2 \) m | \( 3.6 \pm .2 \) m | 1 decimal (tenths place) | | Relative error | \( .06 \pm .02 \) | \( .06 \) | 1–2 significant figures | | Percent relative error | \( 6\% \) | \( 6\% \) | 1 significant figure | --- ## **Extra Notes on Significant Figures** - **Addition/Subtraction:** Use the least number of decimal places from the inputs. - **Error values:** Usually, 1 significant figure. If it starts with 1 or 2, you may use 2 significant figures. - **Final answer:** The value should match the decimal place of the error. Let me know if you want to see each step in more detail!

## Final Solutions Summary ### 1. Final Value with Error - **Value:** \( 3.6 \, \text{m} \) - **Error:** \( \pm 0.2 \, \text{m} \) - **Final Format:** \[ y = 3.6 \pm 0.2 \, \text{m} \] ### 2. Relative Error - **Calculation:** \[ \text{Relative Error} = \frac{0.2}{3.6} \approx 0.056 \] - **Final Result:** \[ 0.06 \pm 0.02 \] ### 3. Percent Relative Error - **Calculation:** \[ \text{Percent Relative Error} = 0.056 \times 100\% \approx 5.6\% \] - **Final Result:** \[ 6\% \] --- ### Significant Figures Summary - **Final Value:** 1 decimal place (tenths) - **Relative Error:** 1 significant figure - **Percent Relative Error:** 1 significant figure Feel free to ask if you need any further clarification!